"Bootstrap, confidence sets, and asymptotic analysis for some procedures of statistical inference"

虽然我的研究计划涉及统计和概率论的许多不同方面，但它围绕着一个主题：渐近（统计和/或分布的扩展）。这项研究包括以下四个主要组成部分： 1.将考虑从两个独立样本的数据中产生的两个二项比例的比值的区间估计问题的一般性陈述。每个样本可以在直接或逆二项抽样的框架中获得。根据不同类型的抽样方案构造了渐近置信区间，并得到了成功概率的无偏估计量及其计算公式。两个二项比例的比率出现在前瞻性研究、生物学实验和工业质量控制生产工艺的比较中。二项比例比率（通常称为相对风险）的统计程序在临床试验和流行病学研究中也很常见，或者更普遍地在制药环境中。因此，我们期望我的这部分（主要是理论）研究将有许多重要的应用。 2.研究了非经典Bootstrap U统计量偏离样本U统计量的渐近概率。我最感兴趣的是这种极限定理的依赖自助随机变量的法律的单一对数，中心极限定理，和法律的大数。 3.将研究基于具有恒定置信水平的收缩估计的置信区域的构造。这将需要修正覆盖概率公式的置信度集中在两个詹姆斯-斯坦估计。 4.将研究裂纹发展的随机模型（与Birnbaum-Saunders和逆高斯分布相关的模型）的构建。将提供这些新模型的参数估计和假设检验的统计问题的解决方案。

Although my research program concerns many different aspects of statistics and probability theory, it revolves around a single theme: asymptotics (expansion of a statistic and/or of a distribution). The research involves the following four main components: 1. A general statement of the interval estimation problem for the ratio of two binomial proportions arising from data coming from two independent samples will be considered. Each sample may be obtained in the framework of direct or inverse binomial sampling. Asymptotic confidence intervals are constructed in accordance with different types of sampling schemes and unbiased estimators of success probabilities and their logarithms will be obtained. The ratio of two binomial proportions arises in prospective studies, in biological experiments, and in the comparison of manufacturing processes for quality control in industry. Statistical procedures for the ratio of binomial proportions (often called the relative risk) are also quite common in clinical trials and epidemiological studies, or more generally in a pharmaceutical setting. Hence we expect that this part of my (mostly theoretical) research will have many significant applications. 2. Asymptotic probability for the deviations of some nonclassical bootstrap U-statistics from the sample U-statistic will be investigated. I am mostly interested in such limit theorems for dependent bootstrap random variables as the law of the single logarithm, the central limit theorem, and the law of large numbers. 3. The construction of a confidence region based on the shrinkage estimator with constant confidence level will be investigated. This will require rectification of formulae for coverage probabilities by confidence sets centered at both James-Stein estimators. 4. Construction of stochastic models for crack development (models related to the Birnbaum-Saunders and Inverse Gaussian distributions) will be investigated. Solutions to the statistical problems of parameter estimation and hypothesis testing for these new models will be provided.